Interestingly, I'm sure the Eaton instructions said mineral oil?
You can use whatever your heart desires..Olive oil if u want,it's your diff not mine,I wouldn't put mineral oil into our customers differential,
Unless it's a single spinner B/W or full spool(smaller size carrier/s more space in the housing)
It's no brainer,,syn. oil outlasts and outperforms min. oil.
In simple terms;;
In small housing,spinning thousand time a minute,churning gears creates lots of aeration and foam/air bubbles,straight into the meshing gears gets compressed(bang) creates heat and wear.
Syn. oil will typically have a film strength 5 to 10 times higher then min. oil(less gear wear),
Syn oil simply more tolerant to extreme heat then petroleum based oils are,
Syn oil flow much much faster and better then mineral oil,therefore your expensive pinion bearings won't be starving for lubrication and your carrier bearings won't be dancing in their cups.
Mineral oil requires more service (2 to 1)intervals,
etc. etc. etc..
OIL, .lubricates it..For quiet and strong performance, diff centre,should be setup properly first...